Random continued fractions: a Markov chain approach

Alok Goswami ()

Economic Theory, 2003, vol. 23, issue 1, 85-105

Abstract: By allowing the numbers appearing in a continued fraction to be random, one gets what are called random continued fractions. Under fairly general conditions, including the case when the random variables are i.i.d. non-negative, random continued fractions converge with probability one. A markovian algorithm seems to play a crucial role in studying the distribution of random continued fractions. This Markov Chain on $S=(0,\infty )$ is generated by iteration of random monotone decreasing maps on S and the connection comes from the fact that the distribution of random continued fraction is obtained as an invariant probability of the Markov Chain. Using the splitting condition, it is shown that the distribution of the Markov Chain converges exponentially fast in the Kolmogorov distance to an unique invariant probability $\pi$ , which is shown to be non-atomic, except in the degenerate case. A sufficient condition is given for the invariant probability $\pi$ to have full support S. In some special cases, the invaraint probability is obtained explicitly and this includes one case when the probability $\pi$ turns out to be a singular non-atomic probability with full support S. Extensions of some results to higher dimensions are also discussed. Copyright Springer-Verlag Berlin/Heidelberg 2003

Keywords: Continued fractions; Gauss map; Markov chains; Singular invariant probability; Splitting condition; Iteration of random maps. (search for similar items in EconPapers)
Date: 2003
References: Add references at CitEc
Citations:

Downloads: (external link)
http://hdl.handle.net/10.1007/s00199-002-0333-4 (text/html)
Access to full text is restricted to subscribers.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:joecth:v:23:y:2003:i:1:p:85-105

Ordering information: This journal article can be ordered from
http://www.springer. ... eory/journal/199/PS2

DOI: 10.1007/s00199-002-0333-4

Access Statistics for this article

Economic Theory is currently edited by Nichoals Yanneils

More articles in Economic Theory from Springer, Society for the Advancement of Economic Theory (SAET) Contact information at EDIRC.
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().